Cracking the Code of Core Math

You know your high school math classes? The ones where the teacher assigns twenty canned problems a night and you churn through them in an hour and hand in the problems on binder paper the very next day? The ones where tests ask you to ‘show your work’ or ‘explain’, and in response you summon a storm of algebra, and amid the tempest of frenetic computation, box your answer to shelter it from the sea of symbolic manipulation?

Hopefully, you’ll be glad to hear that math classes at Mudd are nothing like that. Here is an honest review of core math, starting with the very first math class you’ll take at Mudd, Math 30B. I hope you’ll find it more colorful than what’s listed in the course catalog. Disclaimer: I’ve listed the professors I’ve taken the class with, so my perceptions are of the course as taught by a certain professor.

Math 30 (Calculus), with Professor Orrison:

In the beginning was nothing. On the first day of class, Professor Orrison proclaims: “Let there be numbers!”. By the metaphorical seventh day, by which I mean the end of the semester, Professor Orrison and his revered text (Calculus by Michael Spivak) will have built up the tower of modern mathematics. There will first be integers and mathematical induction. Then polynomials. Then a rigorous introduction to epsilon-delta proofs and the concept of limits. Then a derivation of the Mean Value Theorem, the Chain Rule, and differential/integral calculus in all of its philosophical splendor. Unless you were blessed with an extraordinarily rigorous mathematical education in high school, don’t try and place out of this class. You won’t regret it. Except maybe at 3AM on Monday/Thursday mornings when you’re putting the finishing touches on what may be the some of the first rigorous mathematical proofs you’ve ever written.

From the HMC catalog: “A comprehensive view of the theory and techniques of differential and integral calculus of a single variable; infinite series, including Taylor series and convergence tests. Focus on mathematical reasoning, rigor and proof, including continuity, limits, induction. Introduction to multivariable calculus, including partial derivatives, double and triple integrals.”

Math 35 (Probability and Statistics), with Professor Williams:

A less abstruse, but equally vivid, side of mathematics. While Math 35 may seem to be the ugly duckling of core math—less determinism and more ugly decimals—some of my most memorable mathematical experiences were formed during ‘Prob-stat’. For example, take a room full of 100 students, some of whom have violated the Honor Code. How many violators are there? You could simply tell all the violators in the room to stand up, and count how many people were standing. But what if a few violators didn’t want to reveal themselves? Using statistics you can preserve anonymity while collecting accurate data to get a reasonable estimate of how many people actually violated the Honor Code. In addition to this interesting experiment, we also did a statistical analysis of O-ring failure data from the Challenger Space Shuttle, and how good statistics might have been able to prevent the tragedy.

From the HMC catalog: “Sample spaces, events, axioms for probabilities; conditional probabilities and Bayes’ theorem; random variables and their distributions, discrete and continuous; expected values, means and variances; covariance and correlation; law of large numbers and central limit theorem; point and interval estimation; hypothesis testing; simple linear regression; applications to analyzing real data sets.

Math 40 (Introduction to Linear Algebra), with Professor Karp:

Continuing the legacy of Math 30, this half-semester was spent primarily in proving elegant theorems of linear algebra. You wake up at 3AM and realize that matrix operations can represent not only all the algebra you did in high school, but also that you can encode dynamic population models, differentiation and integration of polynomials, and facial recognition (“eigenfaces”!) with linear algebra. My final project was a realization of Pascal’s Triangle as the power series of an infinite-dimensional matrix. After this class, there is no turning back. You take the red pill – you stay in Wonderland and I show you how deep the rabbit hole goes.

From the HMC Catalog: “Theory and applications of linearity, including: vectors, matrices, systems of linear equations, dot and cross products, determinants, linear transformations in Euclidean space, linear independence, bases, eigenvalues, eigenvectors, and diagonalization.

Math 45 (Introduction to Differential Equations), with Professor Yong:

I have to take a moment to bring Professor Yong’s awesomeness to light here. According to urban legend, the reason why we need to get our course credit overloads approved is due to Prof Yong’s habit of regularly taking 30 credits of classes back in the day when he was a Mudder. In this class you’ll see a lot about springs and masses, and you’ll encounter massive déjà vu when you step out of Mechanics recitation and do the exact same problem in Math 45 ten minutes later. Math 40 and 45 (and 60/65) are where the fast pacing of Mudd core math really shines. Differential equations are crucial to understanding basic physics, and regardless of field, a sense of mathematical maturity makes first-year Mudders exceptionally prepared to do summer research, whether on campus, at an REU, or in industry (more about this later).

Modeling physical systems, first-order ordinary differential equations, existence, uniqueness, and long-term behavior of solutions; bifurcations; approximate solutions; second-order ordinary differential equations and their properties, applications; first-order systems of ordinary differential equations.

Math 60 (Multivariable Calculus), with Professors Karp and Martonosi:

“It’s the same thing as Math 30, you just do everything multiple times”—an apt description of Math 60. Again, core math and core physics are friends—concepts in Math 60 show up in Physics 51—Electromagnetic Theory and Optics.

Review of basic multivariable calculus; optimization and the second derivative test; higher order derivatives and Taylor approximations; line integrals; vector fields, curl, and divergence; Green’s theorem, divergence theorem and Stokes’ theorem, outline of proof and applications.

Math 65 (Differential Equations and Linear Algebra II), with Professors Orrison and Jacobsen:

At the global maximum of core math, your brain is ready to explode, especially if you’ve absorbed Math 60 and 65 over the course of three weeks during Summer Math. You’ll feel like you’ve been in Plato’s Cave all your life. Professor Jacobsen presents his “research”, that is, an elaborate model of Romeo and Juliet, and the concept of love in general, as a dynamical system with a single unstable equilibrium point. “For the record, several of my students are happily married.” -JJ

General vector spaces and linear transformations; change of basis and similarity; generalized eigenvectors; Jordan canonical forms. Applications to linear systems of ordinary differential equations, matrix exponential; nonlinear systems of differential equations; equilibrium points and their stability.

Core Math, Summer Math, and Summer Internships

I’m sure there’s a post dedicated to Summer Math and how amazing it is. In short, Summer Math is an intensive three-week program consisting of Math 60 and 65 offered during the first three weeks of summer so that first-years can pursue other opportunities such as research or an internship during the rest of the summer. Since many professors and employers don’t want to spend half of their summer teaching the required math to interns, it’s a great way for first years to build an advanced mathematical background to land positions normally reserved for upperclassmen. In fact, some employers and professors are willing to bend internship schedules around Summer Math with the promise of getting a student with a strong math background!